• Home
  • Textbooks
  • Statistics
  • Sampling Distributions

Statistics

James T. McClave, Terry T. Sincich

Chapter 6

Sampling Distributions - all with Video Answers

Educators

Chapter Questions

01:43

Problem 1

What is the difference between a population parameter and a sample statistic?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:53

Problem 2

What is a sampling distribution of a sample statistic?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:21

Problem 3

The probability distribution shown here describes a population of measurements that can assume values of 0 , $2,4,$ and $6,$ each of which occurs with the same relative frequency:
\begin{tabular}{l|rrrr}
\hline$x$ & 0 & 2 & 4 & 6 \\
\hline$p(x)$ & $1 / 4$ & $1 / 4$ & $1 / 4$ & $1 / 4$ \\
\hline
\end{tabular}
a. List all the different samples of $n=2$ measurements that can be selected from this population.
b. Calculate the mean of each different sample listed in part a.
c. If a sample of $n=2$ measurements is randomly selected from the population, what is the probability that a specific sample will be selected?
d. Assume that a random sample of $n=2$ measurements is selected from the population. List the different values of $\bar{x}$ found in part $\mathbf{b},$ and find the probability of each. Then give the sampling distribution of the sample mean $\bar{x}$ in tabular form.
e. Construct a probability histogram for the sampling distribution of $\bar{x}$

Akhil Choudhary
Akhil Choudhary
Numerade Educator
03:41

Problem 4

Simulate sampling from the population described in Exercise 6.3 by marking the values of $x,$ one on each of four identical coins (or poker chips, etc.). Place the coins (marked $0,2,4,$ and 6 ) into a bag, randomly select one, and observe its value. Replace this coin, draw a second coin, and observe its value. Finally, calculate the mean $\bar{x}$ for this sample of $n=2$ observations randomly selected from the population (Exercise $6.3,$ part $\mathbf{b}$ ). Replace the coins, mix them, and, using the same procedure, select a sample of $n=2$ observations from the population. Record the numbers and calculate $\bar{x}$ for this sample. Repeat this sampling process until you acquire 100 values of $\bar{x}$. Construct a relative frequency distribution for these 100 sample means. Compare this distribution with the exact sampling distribution of $\bar{x}$ found in part e of Exercise 6.3. [Note: The distribution obtained in this exercise is an approximation to the exact sampling distribution. However, if you were to repeat the sampling procedure, drawing two coins not 100 times, but 10,000 times, then the relative frequency distribution for the 10,000 sample means would be almost identical to the sampling distribution of $\bar{x}$ found in Exercise $6.3,$ part e. $]$

Barath Palanisamy
Barath Palanisamy
Numerade Educator
03:35

Problem 5

Consider the population described by the probability distribution shown below. The random variable $\mathrm{x}$ is observed twice with independent observations.
\begin{tabular}{l|llll}
\hline$x$ & 0 & 1 & 2 & 3 \\
\hline$p(x)$ & .3 & .2 & .3 & .2 \\
\hline
\end{tabular}
\begin{tabular}{cccc}
\hline Sample & Probability & Sample & Probability \\
\hline 0,0 & .09 & 2,0 & .09 \\
0,1 & .06 & 2,1 & .06 \\
0,2 & .09 & 2,2 & .09 \\
0,3 & .06 & 2,3 & .06 \\
1,0 & .06 & 3,0 & .06 \\
1,1 & .04 & 3,1 & .04 \\
1,2 & .06 & 3,2 & .06 \\
1,3 & .04 & 3,3 & .04 \\
& & &
\end{tabular}
a. Complete the sampling distribution table. Type the values of $x$ in ascending order.
b. Construct a probability histogram for the sampling distribution of $\bar{x}$
c. What is the probability that $\bar{x}$ is 3 or larger?
d. Would you expect to observe a value of $\bar{x}$ equal to 3 or larger? Explain.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:33

Problem 6

Refer to Exercise 6.5 and find $E(x)=\mu$. Then use the sampling distribution of $\bar{x}$ found in Exercise 6.5 to find the expected value of $\bar{x}$. Note that $E(\bar{x})=\mu$.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:02

Problem 7

Refer to Exercise $6.5 .$ Assume that a random sample of $n=2$ measurements is randomly selected from the population.
a. List the different values that the sample median $M$ may assume, and find the probability of each. Then give the sampling distribution of the sample median.
b. Construct a probability histogram for the sampling distribution of the sample median, and compare it with the probability histogram for the sample mean (Exercise 6.5, part b).

Tyler Moulton
Tyler Moulton
Numerade Educator
02:18

Problem 8

In Example $6.3,$ we used the computer to generate 1,000 samples, each containing $n=11$ observations, from a uniform distribution over the interval from 150 to $200 .$ Now use the computer to generate 500 samples, each containing $n=15$ observations, from that same population. a. Calculate the sample mean for each sample. To approximate the sampling distribution of $\bar{x},$ construct a relative frequency histogram for the 500 values of $\bar{x}$.
b. Repeat part a for the sample median. Compare this approximate sampling distribution with the approximate sampling distribution of $\bar{x}$ found in part a.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:18

Problem 9

Consider a population that contains values of $x$ equal to $00,01,02,03, \ldots, 96,97,98,99 .$ Assume that these values occur with equal probability. Use the computer to generate 500 samples, each containing $n=20$ measurements, from this population. Calculate the sample mean $\bar{x}$ and sample variance $s^{2}$ for each of the 500 samples.
a. To approximate the sampling distribution of $\bar{x},$ construct a relative frequency histogram for the 500 values of $\bar{x}$
b. Repeat part a for the 500 values of $s^{2}$.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:48

Problem 10

What is a point estimator of a population parameter?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:49

Problem 11

What is the difference between a biased and unbiased estimator?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:49

Problem 12

What is the MVUE for a parameter?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:33

Problem 13

What are the properties of an ideal estimator?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:05

Problem 14

Consider the following probability distribution:
\begin{tabular}{l|ccc}
\hline$x$ & 0 & 1 & 4 \\
\hline$p(x)$ & $1 / 3$ & $1 / 3$ & $1 / 3$ \\
\hline
\end{tabular}
a. Find $\mu$ and $\sigma^{2}$.
b. Find the sampling distribution of the sample mean $\bar{x}$ for a random sample of $n=2$ measurements from this distribution.
c. Show that $\bar{x}$ is an unbiased estimator of $\mu$. [Hint: Show that $E(\bar{x})=\Sigma \bar{x} p(\bar{x})=\mu .]$
d. Find the sampling distribution of the sample variance $s^{2}$ for a random sample of $n=2$ measurements from this distribution.
e. Show that $s^{2}$ is an unbiased estimator for $\sigma^{2}$.

Dominador Tan
Dominador Tan
Numerade Educator
04:44

Problem 15

Consider the following probability distribution:
\begin{tabular}{l|ccc}
\hline$x$ & 2 & 4 & 9 \\
\hline$p(x)$ & $1 / 3$ & $1 / 3$ & $1 / 3$ \\
\hline
\end{tabular}
a. Calculate $\mu$ for this distribution.
b. Find the sampling distribution of the sample mean $\bar{x}$ for a random sample of $n=3$ measurements from this distribution, and show that $\bar{x}$ is an unbiased estimator of $\mu$.
c. Find the sampling distribution of the sample median $M$ for a random sample of $n=3$ measurements from this distribution, and show that the median is a biased estimator of $\mu$.
d. If you wanted to use a sample of three measurements from this population to estimate $\mu,$ which estimator would you use? Why?

Lucas Finney
Lucas Finney
Numerade Educator
04:44

Problem 16

Consider the following probability distribution:
\begin{tabular}{l|ccc}
\hline$x$ & 0 & 1 & 2 \\
\hline$p(x)$ & $1 / 3$ & $1 / 3$ & $1 / 3$ \\
\hline
\end{tabular}
a. Find $\mu$.
b. For a random sample of $n=3$ observations from this distribution, find the sampling distribution of the sample
mean.
c. Find the sampling distribution of the median of a sample of $n=3$ observations from this population.
d. Refer to parts $\mathbf{b}$ and $\mathbf{c},$ and show that both the mean and median are unbiased estimators of $\mu$ for this population.
e. Find the variances of the sampling distributions of the sample mean and the sample median.
f. Which estimator would you use to estimate $\mu$ ? Why?

Lucas Finney
Lucas Finney
Numerade Educator
02:01

Problem 17

Use the computer to generate 750 samples, each containing $n=20$ measurements, from a population that contains values of $x$ equal to $1,2, \ldots, 48,49,50 .$ Assume that these values of $x$ are equally likely. Calculate the sample mean $\bar{x}$ and median $M$ for each sample. Construct relative frequency histograms for the 750 values of $\bar{x}$ and the 750 values of $M$. Use these approximations to the sampling distributions of $\bar{x}$ and $M$ to answer the following questions:
a. Does it appear that $\bar{x}$ and $M$ are unbiased estimators of the population mean? [Note: $\mu=25.5 .]$
b. Which sampling distribution displays greater variation?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:05

Problem 18

Refer to Exercise 6.5
a. Show that $\bar{x}$ is an unbiased estimator of $\mu$.
b. Find $\sigma^{2}$
c. Find the probability that the $\bar{x}$ will fall within $2 \sigma_{\bar{x}} \mu$

Victor Salazar
Victor Salazar
Numerade Educator
04:40

Problem 19

Refer to Exercise 6.5
a. Find the sampling distribution of $s^{2}$.
b. Find the population variance $\sigma^{2}$.
c. Show that $s^{2}$ is an unbiased estimator of $\sigma^{2}$..
d. Find the sampling distribution of the sample standard deviation $s$.
e. Show that $s$ is a biased estimator of $\sigma$.

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
00:53

Problem 20

Refer to Exercise $6.7,$ in which we found the sampling distribution of the sample median. Is the median an unbiased estimator of the population mean $\mu ?$

Kari Hasz
Kari Hasz
Numerade Educator
00:50

Problem 21

What do the symbols $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ represent?

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
00:38

Problem 22

How does the mean of the sampling distribution of $\bar{x}$ relate to the mean of the population from which the sample is selected?

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
01:09

Problem 23

How does the standard deviation of the sampling distribution of $\bar{x}$ relate to the standard deviation of the population from which the sample is selected?

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
00:22

Problem 24

Another name given to the standard deviation of $\bar{x}$ is the __________ (Fill in the blank.)

Christopher Stanley
Christopher Stanley
Numerade Educator
03:06

Problem 25

State the Central Limit Theorem.

Jon Southam
Jon Southam
Numerade Educator
00:44

Problem 26

Will the sampling distribution of $\bar{x}$ always be approximately normally distributed? Explain.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:26

Problem 27

Suppose a random sample of $n$ measurements is selected from a population with mean $\mu=225$ and variance $\sigma^{2}=225 .$ For each of the following values of $n,$ give the mean and standard deviation of the sampling distribution of the sample mean $\bar{x}$.
a. $n=9$
b. $n=49$
c. $n=225$
d. $n=500$
e. $n=1,000$
f. $n=2,500$

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
02:02

Problem 28

Suppose a random sample of $n=16$ measurements is selected from a population with mean $\mu$ and standard deviation $\sigma$. For each of the following values of $\mu$ and $\sigma$, give the values of $\mu_{\bar{x}}$ and $\sigma_{\bar{x}^{-}}$
a. $\mu=12, \sigma=3$
b. $\mu=144, \sigma=16$
c. $\mu=24, \sigma=28$
d. $\mu=12, \sigma=96$

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
03:59

Problem 29

Consider the following probability distribution:
\begin{tabular}{l|llll}
\hline$x$ & 3 & 4 & 6 & 9 \\
\hline$p(x)$ & .3 & .1 & .5 & .1 \\
\hline
\end{tabular}
a. Find $\mu, \sigma^{2},$ and $\sigma$.
b. Find the sampling distribution of $\bar{x}$ for random samples of $n=2$ measurements from this distribution by listing all possible values of $\bar{x},$ and find the probability associated with each.
c. Use the results of part $\mathbf{b}$ to calculate $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$. Confirm that $\mu_{\bar{x}}=\mu$ and that $\sigma_{\bar{x}}=\sigma / \sqrt{n}=\sigma / \sqrt{2}$

Christopher Stanley
Christopher Stanley
Numerade Educator
04:10

Problem 30

A random sample of $n=64$ observations is drawn from a population with a mean equal to 20 and standard deviation equal to 16
a. Give the mean and standard deviation of the (repeated) sampling distribution of $\bar{x}$.
b. Describe the shape of the sampling distribution of $\bar{x}$. Does your answer depend on the sample size?
c. Calculate the standard normal z-score corresponding to a value of $\bar{x}=16$
d. Calculate the standard normal z-score corresponding to $\bar{x}=23 .$
e. Find $P(\bar{x}<16)$.
f. Find $P(\bar{x}>23)$.
g. Find $P(16<\bar{x}<23)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:24

Problem 31

A random sample of $n=100$ observations is selected from a population with $\mu=31$ and $\sigma=25$.
a. Find $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$.
b. Describe the shape of the sampling distribution of $\bar{x}$.
c. Find $P(\bar{x} \geq 28)$.
d. Find $P(22.1 \leq \bar{x} \leq 26.8)$.
e. Find $P(\bar{x} \leq 28.2)$.
f. Find $P(\bar{x} \geq 27.0)$.

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
02:11

Problem 32

A random sample of $n=2,500$ observations is selected from a population with $\mu=500$ and $\sigma=150$.
a. What are the largest and smallest values of $\bar{x}$ that you would expect to see?
b. How far, at the most, would you expect $\bar{x}$ to deviate from $\mu$ ?
c. Did you have to know $\mu$ to answer part $\mathbf{b}$ ? Explain.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:44

Problem 33

Consider a population that contains values of $x$ equal to $0,1,2, \ldots, 97,98,99 .$ Assume that the values of $x$ are equally likely. For each of the following values of $n$, use the computer to generate 500 random samples and calculate $\bar{x}$ for each sample. For each sample size, construct a relative frequency histogram of the 500 values of $\bar{x}$. What changes occur in the histograms as the value of $n$ increases? What similarities exist? Use $n=2, n=10, n=25, n=50,$ and $n=100$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:17

Problem 34

Corporate sustainability of CPA firms. Refer to the Business and Society (Mar. 2011) study on the sustainability behaviors of CPA corporations, Exercise 1.26 (p. 50 ). Recall that corporate sustainability refers to business practices designed around social and environmental considerations. The level of support senior managers have for corporate sustainability was measured quantitatively on a scale ranging from 0 to 160 points, where higher point values indicate a higher level of support for sustainability. The study provided the following information on the distribution of levels of support for sustainability: $\mu=68, \sigma=27$. Now consider a random sample of 30 senior managers and let $\bar{x}$ represent the sample mean level of support.
a. Give the value of $\mu_{\bar{x}},$ the mean of the sampling distribution of $\bar{x},$ and interpret the result.
b. Give the value of $\sigma_{\bar{x}},$ the standard deviation of the sampling distribution of $\bar{x},$ and interpret the result.
c. What does the Central Limit Theorem say about the shape of the sampling distribution of $\bar{x}$ ?
d. Find $P(\bar{x}>65)$.

Christopher Stanley
Christopher Stanley
Numerade Educator
04:01

Problem 35

Voltage sags and swells. Refer to the Electrical Engineering (Vol. 95,2013 ) study of the power quality (sags and swells) of a transformer, Exercise 2.127 (p. 110). For transformers built for heavy industry, the distribution of the number of sags per week has a mean of 353 with a standard deviation of $30 .$ Of interest is $\bar{x},$ the sample mean number of sags per week for a random sample of 45 transformers.
a. Find $E(\bar{x})$ and interpret its value.
b. Find $\operatorname{Var}(\bar{x})$.
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. How likely is it to observe a sample mean number of sags per week that exceeds $400 ?$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:59

Problem 36

Phishing attacks to e-mail accounts. In Exercise 2.47 (p. 81), you lcarncd that phishing dcscribcs an attcmpt to extract personal/financial information from unsuspecting people through fraudulent e-mail. Data from an actual phishing attack against an organization were presented in Chance (Summer 2007 ). The interarrival times, i.e., the time differences (in seconds), for 267 fraud box e-mail notifications were recorded and are saved in the PHISH file. For this exercise, consider these interarrival times to represent the population of interest.
a. In Exercise 2.47 you constructed a histogram for the interarrival times. Describe the shape of the population of interarrival times.
b. Find the mean and standard deviation of the population of interarrival times.
c. Now consider a random sample of $n=40$ interarrival times selected from the population. Describe the shape of the sampling distribution of $\bar{x},$ the sample mean. Theoretically, what are $\mu_{\bar{x}}$ and $\sigma_{\bar{x}} ?$
d. Find $P(\bar{x}<90)$.
e. Use a random number generator to select a random sample of $n=40$ interarrival times from the population, and calculate the value of $\bar{x}$. (Every student in the class should do this.)
f. Refer to part e. Obtain the values of $\bar{x}$ computed by the students and combine them into a single data set. Form a histogram for these values of $\bar{x}$. Is the shape approximately normal?
g. Refer to part $\mathbf{f}$. Find the mean and standard deviation of the $\bar{x}$ -values. Do these values approximate $\mu_{\bar{x}}$ and $\sigma_{\bar{x}},$ respectively?

Dominador Tan
Dominador Tan
Numerade Educator
05:13

Problem 37

Physical activity of obese young adults. In a study on the physical activity of young adults, pediatric researchers measured overall physical activity as the total number of registered movements (counts) over a period of time and then computed the number of counts per minute (cpm) for each subject (International Journal of Obesity, Jan. 2007). The study revealed that the overall physical activity of obese young adults has a mean of $\mu=320 \mathrm{cpm}$ and a standard deviation of $\sigma=100 \mathrm{cpm}$. (In comparison, the mean for young adults of normal weight is $540 \mathrm{cpm}$.) In a random sample of $n=100$ obese young adults, consider the sample mean counts per minute, $\bar{x}$.
a. Describe the sampling distribution of $\bar{x}$.
b. What is the probability that the mean overall physical activity level of the sample is between 300 and $310 \mathrm{cpm} ?$
c. What is the probability that the mean overall physical activity level of the sample is greater than $360 \mathrm{cpm} ?$

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
05:15

Problem 38

Cost of unleaded fuel. According to the American Automobile Association $(\mathrm{AAA}),$ the average cost of a gallon of regular unleaded fuel at gas stations in May 2014 was $\$ 3.65$ (AAA Fuel Gauge Report). Assume that the standard deviation of such costs is $\$ .15 .$ Suppose that a random sample of $n=100$ gas stations is selected from the population and the cost per gallon of regular unleaded fuel is determined for each. Consider $\bar{x},$ the sample mean cost per gallon.
a. Calculate $\mu_{\bar{x}}$ and $\sigma_{\bar{x}^{*}}$
b. What is the approximate probability that the sample has a mean fuel cost between $\$ 3.65$ and $\$ 3.67 ?$
c. What is the approximate probability that the sample has a mean fuel cost that exceeds $\$ 3.67 ?$
d. How would the sampling distribution of $\bar{x}$ change if the sample size $n$ were doubled from 100 to $200 ?$ How do your answers to parts $\mathbf{b}$ and $\mathbf{c}$ change when the sample size is doubled?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:23

Problem 39

Requests to a Web server. In Exercise 5.10 (p. 266) you learned that Brighton Webs LTD modeled the arrival time of requests to a Web server within each hour using a uniform distribution. Specifically, the number of seconds $x$ from the start of the hour that the request is made is uniformly distributed between 0 and 3,600 seconds. In a random sample of $n=75$ Web server requests, let $\bar{x}$ represent the sample mean number of seconds from the start of the hour that the request is made.
a. Find $E(\bar{x})$ and interpret its value.
b. Find $\operatorname{Var}(\bar{x})$.
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. Find the probability that $\bar{x}$ is between 1,900 and 1,950 seconds.
e. Find the probability that $\bar{x}$ exceeds 2,150 seconds..

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:20

Problem 40

Shell lengths of sea turtles. Refer to the Aquatic Biology (Vol. 9,2010 ) study of green sea turtles inhabiting the Grand Cayman South Sound lagoon, Exercise 2.85 (p. 97). Research shows that the curved carapace (shell) lengths of these turtles has a distribution with mean $\mu=50 \mathrm{~cm}$ and standard deviation $\sigma=10 \mathrm{~cm} .$ In the study, $n=76$ green sea turtles were captured from the lagoon; the mean shell length for the sample was $\bar{x}=55.5$ $\mathrm{cm}$. How likely is it to observe a sample mean of $55.5 \mathrm{~cm}$ or larger?

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
04:13

Problem 41

Tomato as a taste modifier. Miraculin is a protein naturally produced in a rare tropical fruit that can convert a sour taste into a sweet taste. Refer to the Plant Science (May 2010 ) investigation of the ability of a hybrid tomato plant to produce miraculin, Exercise 5.38 (p. 279 ). Recall that the amount $x$ of miraculin produced in the plant had a mean of 105.3 micrograms per gram of fresh weight with a standard deviation of $8.0 .$ Consider a random sample of $n=64$ hybrid tomato plants and let $\bar{x}$ represent the sample mean amount of miraculin produced. Would you expect to observe a value of $\bar{x}$ less than 103 micrograms per gram of fresh weight? Explain.

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
05:59

Problem 42

Uranium in the Earth's crust. Refer to the American Mineralogist (Oct. 2009 ) study of the evolution of uranium minerals in the Earth's crust, Exercise 5.9 (p. 266). Recall that researchers estimate that the trace amount of uranium $x$ in reservoirs follows a uniform distribution ranging between 1 and 3 parts per million. In a random sample of $n=60$ reservoirs, let $\bar{x}$ represent the sample mean amount of uranium.
a. Find $E(\bar{x})$ and interpret its value.
b. Find $\operatorname{Var}(\bar{x})$.
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. Find the probability that $\bar{x}$ is between $1.5 \mathrm{ppm}$ and $2.5 \mathrm{ppm} .$
e. Find the probability that $\bar{x}$ exceeds $2.2 \mathrm{ppm}$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:04

Problem 43

Critical part failures in NASCAR vehicles. Refer to The Sport Journal (Winter 2007 ) analysis of critical part failures at NASCAR races, Exercise 5.106 (p. 299 ). Recall that researchers found that the time $x$ (in hours) until the first critical part failure is exponentially distributed with $\mu=.10$ and $\sigma=.10 .$ Now, consider a random sample of $n=50$ NASCAR races and let $\bar{x}$ represent the sample mean time until the first critical part failure.
a. Find $E(\bar{x})$ and $\operatorname{Var}(\bar{x})$.
b. Although $x$ has an exponential distribution, the sampling distribution of $\bar{x}$ is approximately normal. Why?
c. Find the probability that the sample mean time until the first critical part failure exceeds .13 hour.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:57

Problem 44

Motivation of drug dealers. Refer to the Applied Psychology in Criminal Justice (Sept. 2009) investigation of the personality characteristics of drug dealers, Exercise 2.102 (p. 105). Convicted drug dealers were scored on the Wanting Recognition (WR) Scale-a scale that provides a quantitative measure of a person's level of need for approval and sensitivity to social situations. (Higher scores indicate a greater need for approval.) Based on the study results, we can assume that the WR scores for the population of convicted drug dealers have a mean of 40 and a standard deviation of $5 .$ Suppose that in a sample of $n=100$ people, the mean WR scale score is $\bar{x}=42 .$ Is this sample likely to have been selected from the population of convicted drug dealers? Explain.

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
02:36

Problem 45

Is exposure to a chemical in Teflon-coated cookware hazardous? Perfluorooctanoic acid (PFOA) is a chemical used in Teflon $\mathbb{B}$ -coated cookware to prevent food from sticking. The Environmental Protection Agency is investigating the potential risk of PFOA as a cancer-causing agent (Science News Online, August 27,2005$)$. It is known that the blood concentration of PFOA in the general population has a mean of $\mu=6$ parts per billion (ppb) and a standard deviation of $\sigma=10$ ppb. Science News Online reported on tests for PFOA exposure conducted on a sample of 326 people who live near DuPont's Teflon-making Washington (West Virginia) Works facility. a. What is the probability that the average blood concentration of PFOA in the sample is greater than 7.5 ppb? b. The actual study resulted in $\bar{x}=300$ ppb. Use this information to make an inference about the true mean ( $\mu$ ) PFOA concentration for the population that lives near DuPont's Teflon facility.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:31

Problem 46

Characteristics of antiwar demonstrators. Refer to the American Journal of Sociology (Jan. 2014) study of the characteristics of antiwar demonstrators in the United States, Exercise 2.106 (p. 105 ). The researchers found that the mean number of protest organizations joined by antiwar demonstrators over a recent 3 -year period was .90 with a standard deviation of $1.10 .$ Assume that these values represent the true mean $\mu$ and true standard deviation $\sigma$ of the population of all antiwar demonstrators over the 3 years.
a. In a sample of 50 antiwar demonstrators selected from the population, what is the expected value of $\bar{x},$ the sample mean number of protest organizations joined by the demonstrators?
b. Find values of the sample mean, $L$ and $U,$ such that $P(L<\bar{x}<U)=.95$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:17

Problem 47

Hand washing versus hand rubbing. Refer to the British Medical Journal (Aug. 17, 2002) study comparing the effectiveness of hand washing with soap and hand rubbing with alcohol, presented in Exercise 2.107 (p. 106). Health care workers who used hand rubbing had a mean bacterial count of 35 per hand with a standard deviation of $59 .$ Health care workers who used hand washing had a mean bacterial count of 69 per hand with a standard deviation of
106. In a random sample of 50 health care workers, all using the same method of cleaning their hands, the mean bacterial count per hand $(\bar{x})$ is less than $30 .$ Give your opinion on whether this sample of workers used hand rubbing with alcohol or hand washing with soap.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:41

Problem 48

Video game players and divided attention tasks. Human Factors (May 2014) published the results of a study designed to determine whether video game players are better than non-video game players at crossing the street when presented with distractions. Participants (college students) entered a street crossing simulator. The simulator was designed to have cars traveling at various high rates of speed in both directions. During the crossing, the students also performed a memory task as a distraction. The researchers found that students who are video game players took an average of 5.1 seconds to cross the street, with a standard deviation of .8 seconds. Assume that the time, $x,$ to cross the street for the population of video game players has $\mu=5.1$ and $\sigma=.8 .$ Now consider a sample of 30 students and let $\bar{x}$ represent the sample mean time (in seconds) to cross the street in the simulator.
a. Find $P(\bar{x}>5.5)$.
b. The 30 students in the sample are all non-video game players. What inference can you make about $\mu$ and/or $\sigma$ for the population of non-video game players? Explain.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:58

Problem 49

Consider a sampling distribution with $p=.14 .$ and samples of size $n$ each. For each value of $n,$ give the mean and standard deviation of the sampling distribution of the sample proportion, $\hat{p}$.
a. For a random sample of size $n=4,000$.
b. For a random sample of size $n=1,000$.
c. For a random sample of size $n=500$.

Idabelle Cunningham
Idabelle Cunningham
Numerade Educator
02:17

Problem 50

Suppose a random sample of $n=591$ measurements is selected from a binomial population with probability of success $p .$ For each of the given values of $p,$ give the mean and standard deviation of the sampling distribution of the sample proportion, $\hat{p}$.
a. $p=.2$
b. $p=.5$
c. $p=.8$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:58

Problem 51

A random sample of $n=70$ measurements is drawn from a binomial population with probability of success .3.
a. Give the mean and standard deviation of the sampling distribution of the sample proportion, $\hat{p} .$
b. Describe the shape of the sampling distribution of $\hat{p}$.
c. Calculate the standard normal $z$ -score corresponding to a value of $\hat{p}=.33$.
d. Find $P(\hat{p}>.33)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:07

Problem 52

A random sample of $n=250$ measurements is drawn from a binomial population with probability of success $.85 .$
a. Find $E(\hat{p})$ and $\sigma_{\hat{p}}$.
b. Describe the shape of the sampling distribution of $\hat{p}$.
c. Find $P(\hat{p}<.9)$.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:01

Problem 53

A random sample of $n=2,000$ measurements is drawn from a binomial population with probability of success $.25 .$ What are the smallest and largest values of $\hat{p}$ you would expect to see?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:47

Problem 54

Consider a population with values of $x$ equal to 0 or $1 .$ Assume that the values of $x$ are equally likely. For each of the following values of $n,$ use a computer to generate 750 random samples and calculate the sample proportion of 1s observed for each sample. Then construct a histogram of the 750 sample proportions for each sample.
a. $n=5$
b. $n=50$
c. $n=250$
d. Refer to histograms, part a-c. What changes occur as the value of $n$ increases? What similarities exist?

Christopher Stanley
Christopher Stanley
Numerade Educator
03:13

Problem 55

Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414,2010 ) study of the trend in the design of social robots, Exercise $2.7(\mathrm{p} .66) .$ The researchers obtained a random sample of 106 social robots through a Web search and determined the number that were designed with legs, but no wheels. Let $\hat{p}$ represent the sample proportion of social robots designed with legs, but no wheels. Assume that in the population of all social robots, $40 \%$ are designed with legs, but no wheels.
a. Give the mean and standard deviation of the sampling distribution of $\hat{p}$
b. Describe the shape of the sampling distribution of $\hat{p}$.
c. Find $P(\hat{p}>.59)$.
d. Recall that the researchers found that 63 of the 106 robots were built with legs only. Does this result cast doubt on the assumption that $40 \%$ of all social robots are designed with legs, but no wheels? Explain.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:38

Problem 56

Paying for music downloads. According to a recent $P e w$ Internet \& American Life Project Survey (Oct. 2010$), 67 \%$ of adults who use the Internet have paid to download music. In a random sample of $n=1,000$ adults who use the Internet, let $\hat{p}$ represent the proportion who have paid to download music.
a. Find the mean of the sampling distribution of $\hat{p}$.
b. Find the standard deviation of the sampling distribution of $\hat{p}$
c. What does the Central Limit Theorem say about the shape of the sampling distribution of $\hat{p} ?$
d. Compute the probability that $\hat{p}$ is less than .75 .
e. Compute the probability that $\hat{p}$ is greater than .50 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:37

Problem 57

Working on summer vacation. According to an Adweek/ Harris (July 2011) poll of U.S. adults, $35 \%$ do not work during their summer vacation. Assume that the true proportion of all U.S. adults that do not work during summer vacation is $p=.35 .$ Now consider a random sample of 500
U.S. adults.
a. What is the probability that between $50 \%$ and $60 \%$ of the sampled adults do not work during summer vacation?
b. What is the probability that over $40 \%$ of the sampled adults do not work during summer vacation?

Christopher Stanley
Christopher Stanley
Numerade Educator
03:52

Problem 58

Superstitions survey. A Harris (Feb. 2013 ) poll found that one-third of Americans ( $33 \%$ ) believe finding and picking up a penny is good luck. Assume that the true proportion of all Americans who believe finding and picking up a penny is good luck is $p=.33 .$ In a random sample of 1,000 Americans, let $\hat{p}$ represent the proportion who believe finding and picking up a penny is good luck.
a. Describe the properties of the sampling distribution of $\hat{p}$.
b. Find $P(\hat{p}<.40)$. Interpret this result.
c. Find $P(\hat{p}>.30)$. Interpret this result.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:44

Problem 59

Crop damage by wild boars. Refer to the Current Zoology (Apr. 2014 ) study of crop damage incurred by wild boars in southern Italy, Exercise 3.18 (p. 158 ). If crop damage by wild boars occurs, the probability that the type of crop damaged is cereals is .45 , legumes .20 , and orchards .05. In a random sample of 100 crop damage incidents caused by wild boars, what is the probability that at least 50 incidents involve
a. cereal crops?
b. legumes?
c. orchards?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:12

Problem 60

Downloading "apps" to your cell phone. Refer to Exercise 4.27 (p. 221). The study found that $40 \%$ of adult cell phone owners have downloaded an application ("app") to their cell phone. Assume this percentage applies to the population of all adult cell phone owners.
a. In a random sample of 50 adult cell phone owners, how likely is it to find that more than $60 \%$ have downloaded an "app" to their cell phone?
b. Refer to part a. Suppose you observe a sample proportion of .62. What inference can you make about the true proportion of adult cell phone owners who have downloaded an "app"?
c. Suppose the sample of 50 cell phone owners is obtained at a convention for the International Association for the Wireless Telecommunications Industry. How will your answer to part b change, if at all?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:42

Problem 61

Hotel guest satisfaction. Refer to the results of the North American Hotel Guest Satisfaction Index Study referenced in Exercise 4.75 (p. 241). Recall that $66 \%$ of hotel guests were aware of the hotel's "green" conservation program; of these guests, $72 \%$ actually participated in the program by reusing towels and bed linens. In a random sample of 100 hotel guests, find the probability that fewer than 42 were aware of and participated in the hotel's conservation efforts.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:15

Problem 62

Fingerprint expertise. Refer to the Psychological Science (Aug. 2011) study of fingerprint identification, Exercise 4.74 (p. 240). Recall that when presented with prints from the same individual, a fingerprint expert will correctly identify the match $92 \%$ of the time. Consider a forensic database of 1,000 different pairs of fingerprints, where each pair is a match.
a. What proportion of the 1,000 pairs would you expect an expert to correctly identify as a match?
b. What is the probability that an expert will correctly identify fewer than 900 of the fingerprint matches?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:24

Problem 63

True or False. The sample mean, $\bar{x}$, will always be equal to $\mu_{\bar{x}}$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:52

Problem 64

True or False. The sampling distribution of $\bar{x}$ is normally distributed, regardless of the size of the sample $n$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:45

Problem 65

True or False. The standard error of $\bar{x}$ will always be smaller than $\sigma$.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:53

Problem 66

Describe how you could obtain the simulated sampling distribution of a sample statistic.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:21

Problem 67

True or False. The sampling distribution of $\hat{p}$ is normally distributed for large $n$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:53

Problem 68

The standard deviation (or, as it is usually called, the standard error) of the sampling distribution for the sample mean, $\bar{x},$ is equal to the standard deviation of the population from which the sample was selected, divided by the square root of the sample size. That is,
$$
\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}
$$
a. As the sample size is increased, what happens to the standard error of $\bar{x} ?$ Why is this property considered important?
b. Suppose a sample statistic has a standard error that is not a function of the sample size. In other words, the standard error remains constant as $n$ changes. What would this imply about the statistic as an estimator of a population parameter?
c. Suppose another unbiased estimator (call it $A$ ) of the population mean is a sample statistic with a standard error equal to
$$
\sigma_{A}=\frac{\sigma}{\sqrt[3]{n}}
$$
Which of the sample statistics, $\bar{x}$ or $A,$ is preferable as an estimator of the population mean? Why?
d. Suppose that the population standard deviation $\sigma$ is equal to 25 and that the sample size is 64. Calculate the standard errors of $\bar{x}$ and $A$. Assuming that the sampling distribution of $A$ is approximately normal, interpret the standard errors. Why is the assumption of (approximate) normality unnecessary for the sampling distribution of $\bar{x} ?$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:39

Problem 69

Consider a sample statistic $A .$ As with all sample statistics, $A$ is computed by utilizing a specified function (formula) of the sample measurements. (For example, if $A$ were the sample mean, the specified formula would sum the measurements and divide by the number of measurements.)
a. Describe what we mean by the phrase "the sampling distribution of the sample statistic $A$."
b. Suppose $A$ is to be used to estimate a population parameter $\alpha .$ What is meant by the assertion that $A$ is an unbiased estimator of $\alpha ?$
c. Consider another sample statistic, $B$. Assume that $B$ is also an unbiased estimator of the population parameter $\alpha .$ How can we use the sampling distributions of $A$ and $B$ to decide which is the better estimator of $\alpha ?$
d. If the sample sizes on which $A$ and $B$ are based are large, can we apply the Central Limit Theorem and assert that the sampling distributions of $A$ and $B$ are approximately normal? Why or why not?

Barath Palanisamy
Barath Palanisamy
Numerade Educator
04:30

Problem 70

A random sample of 50 observations is to be drawn from a large population of measurements. It is known that $40 \%$ of the measurements in the population are 1 's, $20 \%$ are 2 's, $30 \%$ are 3 's, and $10 \%$ are 4 's.
a. Give the mean and standard deviation of the (repeated) sampling distribution of $\bar{x},$ the sample mean of the 50 observations.
b. Describe the shape of the sampling distribution of $\bar{x}$. Does your answer depend on the sample size?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:44

Problem 71

A random sample of $n=64$ observations is selected from a population with $\mu=22.5$ and $\sigma=2.8$. Approximate each of the following probabilities:
a. $P(\bar{x} \leq 22.5)$
b. $P(\bar{x} \leq 22)$
c. $P(\bar{x} \geq 23.5)$
d. $P(21.9 \leq \bar{x} \leq 22.8)$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:34

Problem 72

A random sample of $n=750$ observations is selected from a binomial population with $p=.45$.
a. Give the mean and standard deviation of the (repeated) sampling distribution of $\hat{p},$ the sample proportion of successes for the 750 observations.
b. Describe the shape of the sampling distribution of $\hat{p}$. Does your answer depend on the sample size?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:49

Problem 73

A random sample of $n=300$ observations is selected from a binomial population with $p=.8$. Approximate each of the following probabilities:
a. $P(\hat{p}<.66)$
b. $P(\hat{p}>.55)$
c. $P(.58<\hat{p}<.62)$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:37

Problem 74

Use a statistical software package to generate 150 random samples of size $n=5$ from a population characterized by a normal probability distribution with a mean of 75 and a standard deviation of $8 .$ Compute $\bar{x}$ for each sample, and plot a frequency distribution for the 150 values of $\bar{x}$. Repeat this process for $n=10,25,50,$ and $75 .$ How does the fact that the sampled population is normal affect the sampling distribution of $\bar{x} ?$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:45

Problem 75

Use a statistical software package to generate $150 \mathrm{ran}-$ dom samples of size $n=5$ from a population characterized by a uniform probability distribution with $c=5$ and $d=20 .$ Compute $\bar{x}$ for each sample, and plot a frequency distribution for the $150 \bar{x}$ values. Repeat this process for $n=10,25,50,$ and $75 .$ Explain how your plots illustrate the Central Limit Theorem.

Christopher Stanley
Christopher Stanley
Numerade Educator
05:08

Problem 76

Suppose $x$ equals the number of heads observed when a single coin is tossed; that is, $x=0$ or $x=1$. The population corresponding to $x$ is the set of 0 's and 1 's generated when the coin is tossed repeatedly a large number of times. Suppose we select $n=4$ observations from this population. (That is, we toss the coin four times and observe 4 values of $x .)$
a. List the different samples (combinations of 0 's and 1 's) that could be obtained.
b. Calculate the value of $\bar{x}$ for each of the samples.
c. List the values that $\bar{x}$ can assume, and find the probabilities of observing these values.
d. Construct a graph of the sampling distribution of $\bar{x}$.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
02:19

Problem 77

A random sample of size $n$ is to be drawn from a large population with mean 150 and standard deviation $20,$ and the sample mean $\bar{x}$ is to be calculated. To see the effect of different sample sizes on the standard deviation of the sampling distribution of $\bar{x},$ plot $\sigma / \sqrt{n}$ against $n$ for $n=1,2,5,10,25,50,$ and 75

Christopher Stanley
Christopher Stanley
Numerade Educator
03:26

Problem 78

Salaries of travel managers. According to a 2012 Business Travel News survey, the average salary of a travel manager is $\$ 110,550 .$ Assume that the standard deviation of such salaries is $\$ 25,000 .$ Consider a random sample of 40 travel managers, and let $\bar{x}$ represent the mean salary for the sample.
a. What is $\mu_{\bar{x}}$ ?
b. What is $\sigma_{\bar{x}}$ ?
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. Find the $z$ -score for the value $\bar{x}=\$ 100,000$.
e. Find $P(\bar{x}>100,000)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:01

Problem 79

Children's attitudes toward reading. In the journal Knowledge Quest (Jan./Feb. 2002), education professors investigated children's attitudes toward reading. One study measured third through sixth graders' attitudes toward recreational reading on a 140 -point scale (where higher scores indicate a more positive attitude). The mean score for this population of children was 106 points, with a standard deviation of 16.4 points. Consider a random sample of 36 children from this population, and let $\bar{x}$ represent the mean recreational reading attitude score for the sample.
a. What is $\mu_{\bar{x}}$ ?
b. What is $\sigma_{\bar{x}}$ ?
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. Find the $z$ -score for the value $\bar{x}=100$ points.
e. Find $P(\bar{x}<100)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:24

Problem 80

Dentists' use of laughing gas. According to the American Dental Association, $60 \%$ of all dentists use nitrous oxide in their practice. In a random sample of 100 dentists, let $\hat{p}$ represent the proportion who use laughing gas in their practice.
a. Find $E(\hat{p})$.
b. Find $\sigma_{\hat{p}}$.
c. Describe the shape of the sampling distribution of $\hat{p}$.
d. Find $P(\hat{p}>.65)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:21

Problem 81

Marital name changing. Refer to the Advances in Applied Sociology (Nov. 2013) study of marital name changing, Exercise 4.78 (p. 241 ). According to the study, the probability that an American female will change her last name upon marriage is .9. Consider a sample of 250 American females. Let $\hat{p}$ represent the proportion of those females who change their last name upon marriage.
a. Find the mean and standard deviation of $\hat{p}$.
b. Calculate the interval, $E(\hat{p}) \pm 2 \sigma_{\hat{p}}$
c. If samples of 250 American females are drawn repeatedly a large number of times and the sample proportion, $\hat{p},$ determined for each sample, what proportion of the $\hat{p}$ values will fall within the interval, part $\mathbf{b} ?$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:06

Problem 82

Violence and stress. Interpersonal violence (e.g., rape) generally leads to psychological stress for the victim. Clinical Psychology Review (Vol. 15,1995 ) reported on the results of all recently published studies of the relationship between interpersonal violence and psychological stress. The distribution of the time elapsed between the violent incident and the initial sign of stress has a mean of 5.1 years and a standard deviation of 6.1 years. Consider a random sample of $n=150$ victims of interpersonal violence. Let $\bar{x}$ represent the mean time elapsed between the violent act and the first sign of stress for the sampled victims.
a. Give the mean and standard deviation of the sampling distribution of $\bar{x}$.
b. Will the sampling distribution of $\bar{x}$ be approximately normal? Explain.
c. Find $P(\bar{x}>5.5)$.
d. Find $P(4<\bar{x}<5)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:21

Problem 83

Research on eating disorders. Refer to The American Statistician (May 2001) study of female students who suffer from bulimia, presented in Exercise 2.44 (p. 80 ). Recall that each student completed a questionnaire from which a "fear of negative evaluation" (FNE) score was produced. (The higher the score, the greater is the fear of negative evaluation.) Suppose the FNE scores of bulimic students have a distribution with mean $\mu=18$ and standard deviation $\sigma=5 .$ Now, consider a random sample of 45 female students with bulimia.
a. What is the probability that the sample mean FNE score is greater than $17.5 ?$
b. What is the probability that the sample mean FNE score is between 18 and $18.5 ?$
c. What is the probability that the sample mean FNE score is less than $18.5 ?$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:17

Problem 84

Producing machine bearings. To determine whether a metal lathe that produces machine bearings is properly adjusted, a random sample of 36 bearings is collected and the diameter of each is measured.
a. If the standard deviation of the diameters of the bearings measured over a long period of time is .005 inch, what is the approximate probability that the mean diameter $\bar{x}$ of the sample of 36 bearings will lie within .0002 inch of the population mean diameter of the bearings?
b. If the population of diameters has an extremely skewed distribution, how will your approximation in part a be affected?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:40

Problem 85

Quality control. Refer to Exercise $6.84 .$ The mean diameter of the bearings produced by the machine is supposed to be .5 inch. The company decides to use the sample mean to decide whether the process is in control (i.e., whether it is producing bearings with a mean diameter of .5 inch $) .$ The machine will be considered out of control if the mean of the sample of $n=36$. Diameters is less than .5025 inch or larger than .5045 inch. If the true mean diameter of the bearings produced by the machine is .505 inch, what is the approximate probability that the test will imply that the process is out of control?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:07

Problem 86

Errors in filling prescriptions A large number of preventable errors (e.g., overdoses, botched operations, misdiagnoses) are being made by doctors and nurses in
U.S. hospitals. A study of a major metropolitan hospital revealed that of every 100 medications prescribed or dispensed, 1 was in error, but only 1 in 500 resulted in an error that caused significant problems for the patient. It is known that the hospital prescribes and dispenses 40,000 medications per year.
a. What is the expected proportion of errors per year at this hospital? The expected proportion of significant errors per year?
b. Within what limits would you expect the proportion of significant errors per year to fall?

Nick Johnson
Nick Johnson
Numerade Educator
02:52

Problem 87

$\begin{array}{lll}\text { Susceptibility to } & \text { hypnosis. } & \text { The } & \text { Computer-Assisted }\end{array}$ Hypnosis Scale (CAHS) is designed to measure a person's susceptibility to hypnosis. CAHS scores range from 0 (no susceptibility) to 12 (extremely high susceptibility). A study in Psychological Assessment (Mar. 1995 ) reported a mean CAHS score of 4.59 and a standard deviation of
2. 95 for University of Tennessee undergraduates. Assume that $\mu=4.59$ and $\sigma=2.95$ for this population. Suppose a psychologist uses the CAHS to test a random sample of 50 subjects.
a. Would you expect to observe a sample mean CAHS score of $\bar{x}=6$ or higher? Explain.
b. Suppose the psychologist actually observes $\bar{x}=6.2 .$ On the basis of your answer to part a, make an inference about the population from which the sample was selected.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 88

Supercooling temperature of frogs. Many species of terrestrial tree frogs can survive prolonged exposure to low winter temperatures during hibernation. In freezing conditions, the frog's body temperature is called its supercooling temperature. A study in Science revealed that the supercooling temperature of terrestrial frogs frozen at $-6^{\circ} \mathrm{C}$ has a relative frequency distribution with a mean of $-2^{\circ} \mathrm{C}$ and a standard deviation of $.3^{\circ} \mathrm{C}$. Consider the mean supercooling temperature $\bar{x}$ of a random sample of $n=42$ terrestrial frogs frozen at $-6^{\circ} \mathrm{C}$
a. Find the probability that $\bar{x}$ exceeds $-2.05^{\circ} \mathrm{C}$.
b. Find the probability that $\bar{x}$ falls between $-2.20^{\circ} \mathrm{C}$ and $-2.10^{\circ} \mathrm{C}$

Christopher Stanley
Christopher Stanley
Numerade Educator
03:17

Problem 89

Levelness of concrete slabs. Geotechnical engineers use water-level "manometer" surveys to assess the levelness of newly constructed concrete slabs. Elevations are typically measured at eight points on the slab; of interest is the maximum differential between elevations. The Journal of Performance of Constructed Facilities (Feb. 2005) published an article on the levelness of slabs in California residential developments. Elevation data collected on over 1,300 concrete slabs before tensioning revealed that the maximum differential $x$ has a mean of $\mu=.53$ inch and a standard deviation of $\sigma=.193$ inch. Consider a sample of $n=50$ slabs selected from those surveyed, and let $\bar{x}$ represent the mean of the sample.
a. Describe fully the sampling distribution of $\bar{x}$.
b. Find $P(\bar{x}>.58)$.
c. The study also revealed that the mean maximum differential of concrete slabs measured after tensioning and loading is $\mu=.58$ inch. Suppose the sample data yield $\bar{x}=.59$ inch. Comment on whether the sample measurements were obtained before tensioning or after tensioning and loading.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:47

Problem 90

Machine repair time. [Note: This exercise refers to an optional section in Chapter 5.] An article in Industrial Engineering (Aug. 1990) discussed the importance of modeling machine downtime correctly in simulation studies. As an illustration, the researcher considered a single-machine-tool system with repair times (in minutes) that can be modeled by an exponential distribution with $\theta=60 .$ (See optional Section $5.6 .$ ) Of interest is the mean repair time $\bar{x}$ of a sample of 100 machine breakdowns.
a. Find $E(\bar{x})$ and the variance of $\bar{x}$.
b. What probability distribution provides the best model of the sampling distribution of $\bar{x}$ ? Why?
c. Calculate the probability that the mean repair time $\bar{x}$ is no longer than 30 minutes.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:38

Problem 91

Flaws in aluminum siding. [Note: This exercise refers to an optional section in Chapter 4.] A building contractor has decided to purchase a load of factory-reject aluminum siding as long as the average number of flaws per piece of siding in a sample of size 45 from the factory's reject pile is 1.5 or less. If it is known that the number of flaws per piece of siding in the factory's reject pile has a Poisson probability distribution with a mean of $1.75,$ find the approximate probability that the contractor will not purchase a load of siding. [Hint: If $x$ is a Poisson random variable with mean $\lambda,$ then $\sigma_{\bar{x}}^{2}$ equals $\left.\lambda / n .\right]$

Christopher Stanley
Christopher Stanley
Numerade Educator
03:03

Problem 92

Soft-drink bottles. A soft-drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least 150 pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of 155 psi and a standard deviation of 2 psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the vendor's production process to verify the vendor's claim. The bottler randomly selects 50 bottles from the last 10,000 produced, measures the internal pressure of each, and finds the mean pressure for the sample to be .75 psi below the process mean cited by the vendor.
a. Assuming the vendor's claim to be true, what is the probability of obtaining a sample mean this far or farther below the process mean? What does your answer suggest about the validity of the vendor's claim?
b. If the process standard deviation were 2 psi as claimed by the vendor, but the mean were 154 psi, would the observed sample result be more or less likely than the result in part a? What if the mean were 156 psi?
c. If the process standard deviation were 1.5 psi as claimed by the vendor, but the mean were 155 psi, would the observed sample result be more or less likely than the result in part a? What if the mean were 3 psi?

Christopher Stanley
Christopher Stanley
Numerade Educator
03:17

Problem 93

Fecal pollution at Huntington Beach. The state of California mandates fecal indicator bacteria monitoring at all public beaches. When the concentration of fecal bacteria in the water exceeds a certain limit ( 400 colony-forming units of fecal coliform per 100 milliliters), local health officials must post a sign (called surf zone posting) warning beachgoers of potential health risks upon entering the water. For fecal bacteria, the state uses a single-sample standard; that is, if the fecal limit is exceeded in a single sample of water, surf zone posting is mandatory. This single-sample standard policy has led to a recent rash of beach closures in California.

Joon Ha Kim and Stanley B. Grant, engineers at the University of California at Irvine, conducted a study of the surf water quality at Huntington Beach in California and reported the results in Environmental Science $\&$ Technology (Sept. 2004 ). The researchers found that beach closings were occurring despite low pollution levels in some instances, while in others, signs were not posted when the fecal limit was exceeded. They attributed these "surf zone posting errors" to the variable nature of water quality in the surf zone (for example, fecal bacteria concentration tends to be higher during ebb tide and at night) and the inherent time delay between when a water sample is collected and when a sign is posted or removed. In order to prevent posting errors, the researchers recommend using an averaging method, rather than a single sample, to determine unsafe water quality. (For example, one simple averaging method is to take a random sample of multiple water specimens and compare the average fecal bacteria level of the sample with the limit of $400 \mathrm{cfu} / 100 \mathrm{~mL}$ in order to determine whether the water is safe.)

Discuss the pros and cons of using the singlesample standard versus the averaging method. Part of your discussion should address the probability of posting a sign when in fact the water is safe and the probability of posting a sign when in fact the water is unsafe. (Assume that the fecal bacteria concentrations of water specimens at Huntington Beach follow an approximately normal distribution.)

Sheryl Ezze
Sheryl Ezze
Numerade Educator